## A bordism approach to string topology.(English)Zbl 1086.55004

Using classical intersection theory of chains in a closed manifold, Chas and Sullivan have defined new operations on the free loop space homology $$H_*(LM)$$ of $$M$$. These operations are called loop product, loop bracket, string bracket, etc... A purely homotopical presentation has been given by R. L. Cohen and J. D. S. Jones using a ring spectrum structure on a Thom spectrum of a virtual bundle over $$LM$$, with, as a corollary, an isomorphism between the loop algebra $$H_*(LM)$$ and the Hochschild cohomology of the cochain algebra on $$M$$, see [Math. Ann. 324, 773–798 (2002; Zbl 1025.55005)].
The author gives a new approach to string topology: He uses the geometric version of singular homology theory introduced by M. Jakob [Manuscr. Math. 96, No. 1, 67–80 (1998; Zbl 0897.55004)]. The main operations of string topology are then redefined and studied in that context. In particular, he extends the Frobenius structure described by Cohen and Godin on string homology to a homological action of the space of Sullivan chord diagrams on the free loop space.

### MSC:

 55P35 Loop spaces 55N35 Other homology theories in algebraic topology

### Keywords:

string topology; free loop space homology

### Citations:

Zbl 1025.55005; Zbl 0897.55004
Full Text: